Question

The average library patron reads 42 books each year, with a standard deviation of 10 books. The library would like to reward its top 25% of patrons, as judged by the number of books read each year.

Use a calculator to find how many books must a library patron read to be eligible for the reward if the library is only examining 100 patrons.

Round x¯ and σx¯ up to the nearest whole number

Answer 1

Distribution of  examining 100 patrons.

µ_X̅ = µ = 42
σ_X̅ = σ / √ (n) = 10/√100 = 1

The library would like to reward its top 25% of patrons.

X ~ N ( µ = 42 , σ = 1 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.25 = 0.75
To find the value of x
Looking for the probability 0.75 in standard normal table to calculate Z score = 0.6745
Z = ( X - µ ) / σ
0.6745 = ( X - 42 ) / 1
X = 42.6745 ≈ 43
P ( X > 43 ) = 0.25

how many books must a library patron read to be eligible for the reward..?

Library patron must read 43 books to eligible for reward.